Mathematical Analysis of Host–Parasitoid Dynamics

Abstract

Host and parasitoid systems are of great interest to ecologists, both because of the global prevalence of insect parasitoids and the impact of parasitoids in regulating their hosts. The direct connection between parasitized hosts and parasitoid offspring leads to simple and specific modeling assumptions. The discrete-time models used for host–parasitoid interactions are also sometimes used for more general predator–prey systems or even to describe consumer–resource dynamics in the broadest sense. In this dissertation, I examine some of the specific building blocks involved in formulation of host–parasitoid models and determine the impacts of these assumptions. I begin with an introduction that includes biological context and a brief overview of the mathematical frameworks used in my work. Next, I present a systematic comparison and analysis of four discrete-time, host–parasitoid models. For each model, I specify that density-dependent effects occur prior to parasitism in the life cycle of the host. I compare density-dependent growth functions arising from the Beverton–Holt and Ricker maps, as well as parasitism functions assuming either a Poisson or negative binomial distribution for parasitoid attacks. I show that overcompensatory density-dependence leads to period-doubling bifurcations, which may be supercritical or subcritical. Stronger parasitism from the Poisson distribution leads to loss of stability of the coexistence equilibrium through a Neimark--Sacker bifurcation, resulting in population cycles. My analytic results also revealed dynamics for one of my models that were previously undetected by authors who conducted a numerical investigation. In this section, I also emphasize the importance of clearly presenting biological assumptions that are inherent to the structure of a discrete-time model in order to promote communication and broader understanding. Climate change has created new and evolving environmental conditions, impacting all species, including hosts and parasitoids. Building on my work with nonspatial host–parasitoid models, I next consider integrodifference equation (IDE) models of host–parasitoid systems to incorporate space and climate-driven shifts in habitats. I describe and analyze three IDE models of host--parasitoid systems to determine criteria for coexistence of the host and parasitoid. Specifically, I determine the critical habitat speed, beyond which the parasitoid cannot survive. By comparing the results from three IDE models, I investigate the impacts of assumptions that reduce the system to a single-species model. I also compare critical speeds predicted by a spatially-implicit difference-equation model with critical speeds determined from numerical simulations of the IDE system. The spatially-implicit model uses approximations for the dominant eigenvalue of an integral operator. The classic methods to approximate the dominant eigenvalue for IDE systems do not perform well for asymmetric kernels, including those that are present in shifting-habitat IDE models. Therefore, I compare several methods for approximating dominant eigenvalues and ultimately conclude that geometric symmetrization and iterated geometric symmetrization give the best estimates of the parasitoid critical speed.